redirect System.out to JTextArea

The following code is meant to redirect text from System.out to a
JTextArea. However, it does not make each output visible immediately:
it sort of buffers. How can I change it, so it shows eacht line
immediately after an "System.out.println(...); ?

======================================
  private void updateTextArea(final String text) {
    SwingUtilities.invokeLater(new Runnable() {
      public void run() {
        textArea.append(text); // it will append, but not show
immediately ...
      }
    });
  }

  private void redirectSystemStreams() {
    OutputStream out = new OutputStream() {
      @Override
      public void write(int b) throws IOException {
        updateTextArea(String.valueOf((char) b));
      }

      @Override
      public void write(byte[] b, int off, int len) throws IOException
{
        updateTextArea(new String(b, off, len));
      }

      @Override
      public void write(byte[] b) throws IOException {
        write(b, 0, b.length);
      }
    };

    System.setOut(new PrintStream(out, true));
    System.setErr(new PrintStream(out, true));
  }